Help with math notation (noob question)

Question

I'm teaching myself math and I can't figure out what the below symbols mean in a textbook:

I did figure out that $\Bbb Z$ = integer set and $\Bbb N$ = natural, and $\Bbb R$ = real. But what do all the pluses and dots and $\{0\}$ mean? Eg how is the first $\Bbb Z$ different from the second?

Broader question - what book / tutorial exists that can help me get comfortable with notation?

Thanks a ton.

Answer 1

• $(\mathbb Z,+)$ is the set $\mathbb Z$ endowed with the operation addition;
• $(\mathbb Z,\cdot)$ is the set $\mathbb Z$ endowed with the operation multiplication;
• $(\mathbb N_0,+)$ is the set $\mathbb N_0$ endowed with the operation addition;
• $\mathbb R\setminus\{0\}=\{x\in\mathbb R\mid x\neq0\}$;
• $(\mathbb R^n,+)$ is the set $\mathbb R^n$ endowed with the operation addition;
• $(\mathbb Z^n,+)$ is the set $\mathbb Z^n$ endowed with the operation addition.

Answer 2

The symbol represent

• $(\mathbb Z,+)$ a group on integer numbers by sum
• $(\mathbb Z,\cdot)$ a group on integer numbers by product
• $(\mathbb N,+)$ a structure on natural numbers by sum
• $\mathbb R \setminus \{0\}$ is the set of real numbers without the zero element (indeed the symbol "$\setminus$" indicates the subtraction for sets)
• $n\in \mathbb N$ means that $n$ belongs to the set of natural numbers